Process control method

1. A process control method for use in a heating apparatus comprising a plurality of heating zones, the method comprising: obtaining a baseline CD (critical dimension) map W by measuring CD at predetermined locations within a substrate processed with the heating apparatus at a baseline setting; calculating an error CD map ΔW by subtracting CD value of a target CD map from that of the baseline CD map at each of the predetermined locations within the substrate; obtaining a CD map W i for each heating zone i by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at a setting where the temperature of the heating zone i is deviated from an original setting by a predetermined value ΔT i ; calculating a basis function P i for each heating zone i by subtracting CD value of an original CD map from that of a perturbed CD map at each of the predetermined locations within the substrate; expanding the error CD map ΔW by a collection of the basis functions with a corresponding expansion coefficient c i for each basis function P i ; and adjusting the baseline setting of each heating zone i of the heating apparatus according to the error CD map ΔW, the expansion coefficient c i , and the basis functions.

2. The method of claim 1 further comprising adjusting the baseline setting of each heating zone i of the heating apparatus by an amount about −c i ΔT i .

3. The method of claim 1 wherein CD is based on an after-developement or after-etching image.

4. The method of claim 1 further comprising obtaining the expansion coefficient c i using a least-square-fitting method.

5. The method of claim 1 further comprising processing a substrate with the heating apparatus at the adjusted baseline setting in semiconductor mask or wafer manufacturing.

6. The method of claim 1 further comprising obtaining the target CD map M by assigning CD value at each of the predetermined locations within a substrate.

7. The method of claim 1 further comprising obtaining the original CD map N by measuring CD at the predetermined locations within a substrate processed with the heating apparatus at the original setting.

8. A heating system comprising a plurality of heating zones, the system comprising: means for obtaining a baseline CD (critical dimension) map W by measuring CD at predetermined locations within a substrate processed with the heating apparatus at a baseline setting; means for calculating an error CD map ΔW by subtracting CD value of a target CD map from that of the baseline CD map at each of the predetermined locations within the substrate; means for obtaining a perturbed CD map W i for each heating zone i by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at a setting where the temperature of the heating zone i is deviated from an original setting by a predetermined value ΔT i ; means for calculating a basis function P i for each heating zone i by subtracting CD value of an original CD map from that of the perturbed CD map at each of the predetermined locations within the substrate; means for expanding the error CD map ΔW by a collection of the basis functions with a corresponding expansion coefficient c i for each basis function P i ; and means for adjusting the baseline setting of each heating zone i of the heating apparatus according to the error CD map ΔW, the expansion coefficient c i , and the basis functions.

9. The heating system of claim 8 further comprising means for adjusting the baseline setting of each heating zone i of the heating apparatus by an amount about −c i ΔT i .

10. The heating system of claim 8 wherein CD is based on an after-developement or after-etching image.

11. The heating system of claim 8 wherein the expansion coefficient c i is obtained using a least-square-fitting method.

12. The heating system of claim 8 further comprising means for processing a substrate with the heating system at the adjusted baseline setting in semiconductor mask or wafer manufacturing.

13. The heating system of claim 8 wherein the target CD map M is obtained by assigning CD value at each of the predetermined locations within a substrate.

14. The heating system of claim 8 wherein the original CD map N is obtained by measuring CD at the predetermined locations within a substrate processed with the heating apparatus at the original setting.

15. A machine-readable storage medium storing a computer program, causing, when executed, a computer to perform a process control method, comprising: obtaining a baseline CD (critical dimension) map W by measuring CD at predetermined locations within a substrate processed with the heating apparatus at a baseline setting; calculating an error CD map ΔW by subtracting CD value of a target CD map from that of the baseline CD map at each of the predetermined locations within the substrate; obtaining a perturbed CD map W i for each heating zone i by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at a setting where the temperature of the heating zone i is deviated from an original setting by a predetermined value ΔT i ; calculating a basis function P i for each heating zone i by subtracting CD value of an original CD map from that of the perturbed CD map at each of the predetermined locations within the substrate; expanding the error CD map ΔW by a collection of the basis functions with a corresponding expansion coefficient c i for each basis function P i ; and adjusting the baseline setting of each heating zone i of the heating apparatus according to the error CD map ΔW, the expansion coefficient c i , and the basis functions.

16. A semiconductor fabrication method incorporating a heating apparatus, comprising: heating an exposed photoresist on a substrate on the heating apparatus at a baseline setting; developing the exposed and heated photoresist to delineate a collection of patterns at predetermined locations; measuring CD (critical dimension) of the collection of patterns to obtain a baseline CD map W; assigning CD value at each of the predetermined locations to generate a target CD map M; adjusting temperature distribution of the heating apparatus according to an error CD map ΔW defined as the difference between the baseline CD map W and the target CD map M; dividing the heating apparatus into a plurality of heating zones; obtaining an original CD map N by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at an original setting; obtaining a perturbed CD map W i for each heating zone i by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at a setting where the temperature of the heating zone i is deviated from the original setting by a predetermined value ΔT i ; calculating a basis function P i for each heating zone i by subtracting CD value of the original CD map from that of the perturbed CD map at each of the predetermined locations within the substrate; expanding the error CD map ΔW by a collection of the basis functions with a corresponding expansion coefficient c i for each basis function P i ; and adjusting the baseline setting of each heating zone i of the heating apparatus according to the error CD map ΔW, the expansion coefficient c i , and the basis functions.

17. The method of claim 16 further comprising adjusting the baseline setting of each heating zone i of the heating apparatus by an amount about −c i ΔT i .

18. The method of claim 16 further comprising obtaining the expansion coefficient c i using a least-square-fitting method.

19. The method of claim 16 wherein CD is based on an after-development or after-etching image.

20. The method of claim 16 further comprising processing a substrate by the heating apparatus at the adjusted baseline setting in semiconductor mask or wafer manufacturing.

21. A semiconductor fabrication method incorporating a heating apparatus, comprising: heating an exposed photoresist on a substrate on the heating apparatus at a baseline setting; developing and etching the exposed and heated photoresist to delineate a collection of patterns at predetermined locations; measuring CD (critical dimension) of the collection of patterns to obtain a baseline CD map W; assigning CD value at each of the predetermined locations to generate a target CD map M; adjusting temperature distribution of the heating apparatus according to an error CD map ΔW defined as the difference between the baseline CD map W and the target CD map M; dividing the heating apparatus into a plurality of heating zones; obtaining an original CD map N by measuring CD at the predetermined locations within the substrate processed with the heating appparatus at an original setting; obtaining a perturbed CD map W i for each heating zone i by measuring CD at the predetermined locations within the substrate processed with the heating apparatus at a setting where the temperature of the heating zone i is deviated from the original setting by a predetermined value ΔT i ; calculating a basis function P i for each heating zone i by subtracting CD value of the original CD map from that of the perturbed CD map at each of the predetermined locations within the substrate; expanding the error CD map ΔW by a collection of the basis functions with a corresponding expansion coefficient c i for each basis function P i ; and adjusting the baseline setting of each heating zone i of the heating apparatus according to the error CD map ΔW, the expansion coefficient c i , and the basis functions.

22. The method of claim 21 further comprising adjusting the baseline setting of each heating zone i of the heating apparatus by an amount about −c i ΔT i .

23. The method of claim 21 further comprising obtaining the expansion coefficient c i using a least-square-fitting method.

24. The method of claim 21 wherein CD is based on an after-developement or after-etching image.

25. The method of claim 21 further comprising processing a substrate by the heating apparatus at the adjusted baseline setting in semiconductor mask or wafer manufacturing.